Level 1

Varieties (ignoring reinforcement learning)

Supervised: classification

Supervised: regression

Unsupervised: data

Unsupervised: example result

Level 1: summary

Machine learning comes in two varieties:

• supervised learning:
  • typically lots of data-label pairs
  • aim is to build a model data -> label
  • categorical labels: classification
  • numeric labels: regression
• unsupervised learning:
  • unlabelled data
  • goals are vaguer but generally aims to simplify data and uncover patterns

Level 2

How does a computer “see” a cat?

How many images are possible?

• for a 20 x 20 binary image -> \(X\) has dimensionality of 400
• \(2^{400}\approx 2 \times 10^{120}\) possible images
• a very small proportion of those correspond to real world type images
• a very small proportion of real world images correspond to cats
• idea: even if dimensionality is big, effective dimensionality much lower
  • ML aims to find these lower dimensional representations

Supervised learning

Supervised learning

Rule determination

• Want to learn a rule \(f: X \rightarrow y\)
• Function may be complex and is controlled by low-dimensional parameters: \(f=f(\theta)\)
• Have training data:

\[(X_1, y_1), (X_2, y_2), ..., (X_n, y_n)\]

• Can we learn \(f\) by optimising \(\theta\) on training data?

What is \(\; f\)?

• Linear combination of elements of \(X\) (linear regression)
• Linear combination of functions of elements of \(X\) (kernel regression)
• Regression trees (random forests, boosted regression)
• Non-linear combinations of elements, stacked into multiple layers (deep learning)

How to learn \(\; f\)?

Define a cost of mismatch between \(y_{i}^{\text{pred}}=f(X_i|\theta)\) and \(y_{i}^\text{actual}\):

\[\begin{equation} C(f(X_i|\theta), \;y_{i}^\text{actual}) \geq 0. \end{equation}\]

For cat / dog example, we could do something like: - \(C(1, 0) = C(0, 1) = 1\) - \(C(0, 0) = C(1, 1) = 0\)

Then choose \(\theta\) to minimise costs.

Unsupervised learning

Unsupervised learning

Unsupervised learning: what does \(Z\) capture?

Unsupervised learning: clustering

Level 2: summary

• ML algorithms take numeric objects (vectors / matrices / tensors) as input
• intrinsic dimensionality of most things \(<\) raw dimensions: world simpler
• supervised learning:
  • determines a mathematical function to predict outputs from inputs
  • function depends on parameters which must be learned using training / testing data
  • learning based on optimising cost function

Level 2: summary

• unsupervised learning:
  • attempts to find more parsimonious representation of data
  • low dimensional variables learned may be more interpretable
  • clustering is an example of unsupervised ML

Questions?

Example data

Non-probabilistic model

\[\begin{equation} y_i = \alpha + \beta x_i + \epsilon_i \end{equation}\]

where \(\epsilon_i\) is an error term. Define mean-squared loss:

\[\begin{equation} L = \frac{1}{K} \sum_{i=1}^{K} (y_i - (\alpha + \beta x_i))^2 \end{equation}\]

What does this model look like?

Learning parameters

determine \(\hat{\alpha}\) and \(\hat{\beta}\) as those minimising \(L\):

\[\begin{align} \frac{\partial L}{\partial \alpha} &= -\frac{2}{K}\sum_{i=1}^{K} (y_i - (\alpha + \beta x_i)) = 0\\ \frac{\partial L}{\partial \beta} &= -\frac{2}{K}\sum_{i=1}^{K} x_i (y_i - (\alpha + \beta x_i)) = 0 \end{align}\]

Gradient descent

although a closed form expression exists for \(\hat{\alpha}\) and \(\hat{\beta}\), for more general models, one doesn’t exist \(\implies\) use gradient descent optimisation

1. initialise parameters \(\alpha=\alpha_0\), \(\beta=\beta_0\)
2. in each epoch update parameters:

\[\begin{align} \alpha &= \alpha - \eta \frac{\partial L}{\partial \alpha}\\ \beta &= \beta - \eta \frac{\partial L}{\partial \beta} \end{align}\]

until \(\alpha\) and \(\beta\) no longer change. \(\eta\) is the learning rate

Linear regression with polynomial terms

\[\begin{equation} y_i = \theta_0 + \theta_1 x_i + \theta_2 x_i^2 + ... + \theta_p x_i^p + \epsilon_i \end{equation}\]

model is better able to fit more complex datasets

Is this a good fit?

What went wrong?

• adding more parameters always reduces error on training set
• but results in a model that generalises poorly

What is a good fitting model?

Solutions

• need a separate validation set to test model predictions on
• regularisation can help too, yielding a new objective function:

\[\begin{equation} L = C||\theta||_q + \frac{1}{K} \sum_{i=1}^{K} (y_i - f_p(x_i))^2 \end{equation}\]

where \(||.||_q\) denotes the \(L_q\) norm: different choices can yield very different estimates

Linear regression summary

• linear regression defines a loss function (typically mean squared error) between actual and predicted observations
• training can be done via gradient descent: each epoch corresponds to a single parameter update
• (gradient descent also used to train many other methods, like neural nets)
• fitting regression with more complex functional forms can fit more complex data
• but risks poor generalisation

Questions?

KNN

• non-model-based learning algorithm
• keeps training data in memory when making predictions, in contrast to most other methods
• generally fast to run compared to many model-based approaches
• straightforward premise for both classification and regression

KNN classification

for new data point \(\tilde x_i\):

1. find \(k\) nearest \(x\) values from training data \((x_i, y_i)\) for \(i=1,...,n\)
2. tally up the corresponding \(y\) labels: \((y_1,..,y_k)\)
3. classify \(\tilde x_i \rightarrow \text{mode}(y_1,..,y_k)\)

KNN regression

for new data point \(\tilde x_i\):

1. find \(k\) nearest \(x\) values from training data \((x_i, y_i)\) for \(i=1,...,n\)
2. tally up the corresponding \(y\) labels: \((y_1,..,y_k)\)
3. classify \(\tilde x_i \rightarrow \text{mean}(y_1,..,y_k)\)

Distance metrics

many options possible. Common metrics include:

• Euclidean: \(s(x_1,x_2) = \sqrt{\sum_{i=1}^{D} (x_{1,i} - x_{2,i})^2}\)
• cosine similarity:

\[\begin{equation} s(x_1,x_2) = \frac{x_1.x_2}{|x_1||x_2|} \end{equation}\]

Example data

KNN with k=10

KNN with k=1

KNN with k=50

Issue with KNN

assume

\[\begin{equation} \boldsymbol{x} \sim \mathcal{N}(0, I) \end{equation}\]

where \(I\in\mathbb{R}^d\). What does the distribution of Euclidean distances between points look like as \(d\) changes?

Distance dists: neighbours not near in higher D

KNN summary

• KNN uses a simple device to do both classification and regression
• in both cases taken the \(k\) closest points to choose label or value
• \(k\) controls the smoothness of predictions
• generally cheaper to run than other models
• in higher dimensions, less useful

Questions?

Logistic regression

• confusingly, this is a classifier not a regression (in the ML sense)
• models data as generated from a Bernoulli probability distribution
• probability parameter of Bernoulli modelled by logistic function, hence the name
• simple classifier but yields interpretable results and can be estimated in Bayesian framework

Model for binary data

• suppose we have many labelled tuples of \((x_{i}, y_i)\)
• where \(y_i\) is binary: here, we set \(y_i=0\) for one category; \(y_i=1\) for the other
• since outcome is binary \(\implies\) use an appropriate probability distribution:

\[\begin{equation} y_i \sim \text{Bernoulli}(\theta_i) \end{equation}\]

where \(0\leq \theta_i \leq 1 = Pr(y_i=1)\)

Bernoulli probability distribution

is given by:

\[\begin{equation} \text{Pr}(y_i|\theta_i) = \theta_i^{y_i} (1 - \theta_i)^{1 - y_i} \end{equation}\]

so that \(\text{Pr}(y_i=1) = \theta_i\) and \(\text{Pr}(y_i=0) = 1 - \theta_i\)

Logistic function

In logistic regression, we use logistic function:

\[\begin{equation} \theta_i = f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_i))} \end{equation}\]

Likelihood and Bayesian estimation

assume data are i.i.d., the likelihood is:

\[\begin{equation} L=p(\boldsymbol{y}|\beta,\boldsymbol{x}) = \prod_{i=1}^{K} f_\beta(x_i)^{y_i} (1 - f_\beta(x_i))^{1 - y_i}. \end{equation}\]

Can use gradient descent to find maximum likelihood estimates (or estimate using Bayesian inference).

Multivariate logistic regression

straightforward to extend the model to incorporate multiple regressions:

\[\begin{equation} f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))} \end{equation}\]

But how to interpret parameters of logistic regression?

Log-odds ratios

another way of writing logistic function:

\[\begin{align} f_\beta(x_i) &= \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))}\\ &= \frac{\exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]

so that

\[\begin{align} 1 - f_\beta(x_i) = \frac{1}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]

Log-odds ratios

taking the ratio:

\[\begin{equation} \text{odds} = \frac{f_\beta(x_i)}{1-f_\beta(x_i)} = \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}) \end{equation}\]

so that

\[\begin{equation} \log\text{odds} =\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i} \end{equation}\]

meaning (say) \(\beta_1\) represents the change in log-odds for a one unit change in \(x_{1}\)

Logistic regression summary

• logistic regression models are binary classifiers (in ML speak)
• assumes Bernoulli distribution for outputs
• logistic function used to relate changes in inputs to outputs
• estimatable via Bayesian inference
• multivariate logistic regression is a commonly used tool

Questions?

Summary

• unsupervised and supervised learning aim to achieve different goals
• dimensionality reduction is one variety of unsupervised learning
• PCA is a linear projection method

Summary

• clustering methods can reduce data down to a single dimension
• k-means works for simple datasets

Summary

• supervised ML uses labelled data to make predictions
• linear regression is a form of regression model
• model with more parameters are likely to overfit more
• logistic regression is a binary classifier

How to learn more?

All available on SOLO:

• “The hundred-page machine learning book”, Burkov
• “Hands-On machine learning with Scikit-Learn & Tensorflow”, Geron

Coursera:

• Data Science: Statistics and Machine Learning Specialization, Johns Hopkins